Finite Group Databasebeta

← #18220

Subgroups — #18220

Order 256.

479 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 479 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3502C211yesmaximal, characteristic, complemented
2#3101C212yesmaximal
3#2536C211yesmaximal, characteristic
4#1611C214yesmaximal, complemented
5#1599C214yesmaximal
6#1518C212yesmaximal, complemented
7#1461C211yesmaximal, characteristic, complemented
8Q82—22no—
9#557—22no—
10#557—24no—
11#544—21no—
12#544C2214yes—
13#533—22no—
14#533C2212yes—
15C2×C22⋊Q8—21no—
16#519—21no—
17C2×C4×Q8C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19#483C2214yes—
20#476—22no—
21#461—22no—
22#450C2214yes—
23C4×SD16—24no—
24#431C2212yes—
25#398C2212yes—
26#394C2212yes—
27#388—22no—
28#382C2211yescharacteristic
29#380—21no—
30#328C2218yes—
31#324C2214yes—
32C22×Q8D411yescharacteristic
33C22×D4C2311yescharacteristic
34C23×C4—21no—
35C2×SD16—24no—
36C4⋊Q8—22no—
37C4⋊Q8—22no—
38C4⋊Q8—24no—
39C4⋊Q8—28no—
40C4⋊Q8D412yes—
41C4⋊Q8D412yes—
42#125—24no—
43#125—24no—
44#125—28no—
45#125—22no—
46C22⋊Q8—22no—
47C22⋊Q8—24no—
48C22⋊Q8—44no—
49C22⋊Q8C2314yes—
50C4×Q8—24no—
51C4×Q8—24no—
52C4×Q8—24no—
53C4×Q8—24no—
54C4×Q8—22no—
55C4×Q8—22no—
56C4×Q8—22no—
57C4×Q8D414yes—
58C4×D4—22no—
59C4×D4—22no—
60#117—24no—
61#117—22no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4D411yescharacteristic
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2312yes—
67C2×C22⋊C4—21no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—22no—
70C2×C42D412yes—
71C2×C42C2311yescharacteristic
72#106—24no—
73C4⋊C8—22no—
74#103—24no—
75#102—24no—
76C22⋊C8C2314yes—
77C4×C8—22no—
78#95—22no—
79#95—21no—
80#95C2312yes—
81C24—21no—
82C2×Q8—24no—
83C2×Q8—22no—
84C2×Q8—22no—
85C2×Q8—22no—
86C2×Q8SD1614yescomplemented
87C2×Q8C2×D414yes—
88C2×D4—22no—
89C2×D4—42no—
90C22×C4—22no—
91C22×C4—21no—
92C22×C4—42no—
93C22×C4—42no—
94C22×C4—42no—
95C22×C4C2×D412yes—
96C22×C4C2×D412yes—
97C22×C4C2×D411yescharacteristic
98C22×C4C2×D411yescharacteristic
99C22×C4C2411yescharacteristic
100SD16—48no—
101C2×C8—24no—
102C4⋊C4—24no—
103C4⋊C4—24no—
104C4⋊C4—24no—
105C4⋊C4—28no—
106C4⋊C4—22no—
107C4⋊C4—22no—
108C4⋊C4—22no—
109C4⋊C4—22no—
110C4⋊C4—24no—
111C4⋊C4—24no—
112C4⋊C4—24no—
113C4⋊C4—22no—
114C4⋊C4—42no—
115C4⋊C4—44no—
116C4⋊C4C2×D414yes—
117C22⋊C4—24no—
118C22⋊C4—44no—
119C22⋊C4—41no—
120C22⋊C4—41no—
121C22⋊C4—44no—
122C42—28no—
123C42—22no—
124C42—22no—
125C42—24no—
126C42C2×D412yes—
127C42C2×D412yes—
128C23—22no—
129C23—41no—
130C23—41no—
131C23—41no—
132C23—41no—
133C23—41no—
134C23—41no—
135C23C22×D411yescharacteristic
136Q8—28no—
137Q8—44no—
138Q8—44no—
139Q8—84no—
140D4—24no—
141D4—42no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—24no—
145C2×C4—24no—
146C2×C4—22no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—41no—
153C2×C4—42no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—84no—
157C2×C4#12011yescharacteristic
158C2×C4#12011yescharacteristic
159C2×C4#12012yes—
160C2×C4C2×SD1614yes—
161C2×C4C2×SD1612yes—
162C2×C4#13712yes—
163C2×C4C22×D412yes—
164C8—44no—
165C22—21no—
166C22—21no—
167C22—42no—
168C22—42no—
169C22—42no—
170C22—81no—
171C22—81no—
172C22—81no—
173C22—81no—
174C22C2×#12011yescharacteristic
175C22C22×SD1611yescharacteristic
176C22C2×#13711yescharacteristic
177C4—24no—
178C4—21no—
179C4—21no—
180C4—22no—
181C4—44no—
182C4—42no—
183C4—44no—
184C4—82no—
185C4—81no—
186C4#44814yes—
187C2—21no—
188C2—21no—
189C2—82no—
190C2C2×#44811yescharacteristic
191C2#302011yescharacteristic
192C2#302511yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #3502C2semidirect
4. #1611C2semidirect
6. #1518C2semidirect
7. #1461C2semidirect
86. C2×Q8SD16semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#35022
#31012
#25362
#16112
#15992
#15182
#14612